general topology – When is a P-space the set of P-points of a compact Hausdorff space?

Let $X$ be a Hausdorff space. A point $xin X$ is a P-point if every countable intersection of open neighbourhoods of $x$ is open, and $X$ is a
P-space if every point is P-point, equivalently if every countable intersection of open sets is open.

The set of P-points in a compact Hausdorff space is thus a completely regular P-space. My question is: is every completely regular P-space equal to the set of P-points of some compact Hausdorff space?

If not, is there a characterisation of those completely regular P-spaces which are equal to the set of P-points of some compact Hausdorff space?

Example: a countable discrete set $N$ is a P-space and is equal to the set of P-points of its one-point compactification $Ncup{infty}$.

real analysis – An example that the sum of two Borel sets which is not a Borel set in n-dimensional Euclidean space

By sum of two sets I mean $A+B := {x+y:x in A quad y in B}$, and there is a tip in a book of real analysis by Zhou Minqiang which says:

“If $A,B$ are Borel sets in $mathbb{R}^{n}$, $A+B$ may not be a Borel set.”

I want to know some specific examples.(Maybe $mathbb{R}^{1}$ ?)

Any comments will be helpful.

time complexity – A genral turing model with one tape to define sublinear space (L,NL,..)

A genral turing model with one tape to define sublinear space (L,NL,..)

Normally to define sub-linear space complexity we need special Turing models with many tapes, at least two: a read-only tape and a work tape; or often three: with an additional output tape. (We do this because the reading of the input would cost us linear space.)

I’m looking for a general Turing model with just one tape and one consistent definition for space and for complexity that works in all the cases, inclusively the sub-linear case of L (Logspace), NL (NLogspace).. and the more famous P, NP, PSPACE etc.

For this purpose obviously we must change the space definition – but how best?

Want to upgrade to Mac OS High Sierra and it shows MediaKit reports not enough space on device for requested operation

Trying to update my MacBook from os El Capitan to High Sierra on reboot it shows MediaKit reports not enough space on device for requested operation

How much disk space is required to install KVM hypervisor on a CentOS machine?

I want to install KVM on my CentOS machine. But the main partition size of my CentOS is about 50GB with only 15GB available. My question is should i extend the main partition in order to install KVM ? if yes, how to extend the main partition size?

Thank you.

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ap.analysis of pdes – Boundedness of Riesz potential on hardy space

I encounter the following claim in one paper:

If $(-Delta)^{frac14}uin L^{2,infty}(mathbb{R})$, then $uin BMO(mathbb{R})$. Equivalently, if $uin mathcal{H}^1(mathbb{R})$, then $(-Delta)^{-frac14}uin L^{2,1}(mathbb{R})$. Here $L^{2,infty}$ and $L^{2,1}$ are Lorentz space and $mathcal{H}$ is the hardy space.

I do not know how to show this fact. My knowledge of Riesz potential tells me if $uin mathcal{H}^1(mathbb{R})$, then $(-Delta)^{-frac14}u=I_{1/2}uin L^2(mathbb{R})$, but why does it lie in the smaller space $L^{2,1}$?

The paper says the first half of the claim is contained in the paper: Adams, D. R. (1975). A note on riesz potentials. Duke Mathematical Journal. I read Adams’ paper and could not figure out why.